The visitors, who went down to 10 men in the first half, had equalised through Enoch Showunmi, but Harry Maguire's header handed David Weir's side all three points
Goals from Kevin McDonald and Harry Maguire sandwiched an equaliser from Enoch Showunmi to earn United three points in their first game of the season at Bramall Lane.
Weir's men started the match brightly and were well on top before County's task was made even harder when Gary Liddle was dismissed for a reckless challenge on McDonald after 13 minutes.
The defender lunged for the ball and ended up catching the 24-year-old in the ribs, leaving referee Keith Stroud with no choice but to issue the red card.
This numerical advantage only served to increase the home team's dominance and they had a good shout for a penalty rejected soon after. Striker Chris Porter appeared to be dragged down in the box by County defender Dean Leacock, but Stroud waved away their appeals.
New signing Febian Brandy, a constant menace throughout the game, then fired wide after a cushioned knock-down from Porter, but United did not have to wait long to take a fully-deserved lead.
Right-back Darryl Westlake swung in a dangerous first-time cross and midfielder McDonald - who spent a short time on loan at Meadow Lane in 2011 - timed his run perfectly to place his shot beyond Bartosz Bialkowski on the half-hour mark.
The home side continued to press but County held out to go into the break just one goal down.
And the visitors got themselves back into the game against the run of play shortly after the restart when Showunmi arrowed a 20-yard drive into the bottom corner just two minutes after being brought on by County manager Chris Kiwomya.
However, their parity lasted just eight minutes as United centre-back Maguire powered a header past Bialkowski from Steven McGinn's free-kick.
The home side continued to control the game and on-loan Liverpool midfielder Conor Coady almost got his first goal for the club after coming off the bench, but he could only fire wide after surging into the box.